When is it dangerous to simultaneously protect a SQL database with multiple backup products (such as SQL Maintenance Plans, BackupExec and Unitrends Backup Professional)?

It may be dangerous to your budget to waste money on unnecessary, redundant commercial products.

Seriously, this is a bizarre question. You're beginning with the premise that it's "dangerous to... protect a SQL database" with various backup products. For starters, where did you get that idea?

To address a couple of questions, it sounds like you don't trust any one of those; and, yet, you are worried that if you try to use more than one, they may collide in some way that causes them not to work. The danger to your budget mentioned by anomie is yet another fear.

So, why not use a reliable open source solution that is documented and you can see how it works? Download the community edition of Zmanda Recovery Manager for MySQL. You can use that to do both incremental and full backups of MySQL databases using standard MySQL tools. You can gain an understanding of what those tools are by looking at the Documentation for Database Backup Methods.

You should gain familiarity with the tools you are using and confirm for yourself that they are working. Test them. Watch the files as you are using them. Test recovering a database with them. You will only trust tools when you are familiar with them and know for yourself that they are working.